Abstract : This project was concerned with the problem of determining the physical and geometric properties of an unknown inhomogeneity from a knowledge of its effect on a given time harmonic electromagnetic wave with particular concern to non-destructive testing and medical imaging. The main accomplishments were: (1) The discovery of a linear method for determining the support of aberrant inhomogeneities without any a priori assumptions on either the frequency or magnitude of the inhomogeneity; (2) The application of this new linear method to problems in microwave medical imaging; (3) The analysis and numerical implementation of a method of perfectly matched lay for the solution of Maxwell's equations; and (4) The derivation of an adaptive method for mesh refinement to produce a far field pattern of prescribed accuracy.